If 54 J of work are needed to stretch a spring from 15 cm to 21 cm and 90 J are needed to stretch it from 21 cm to 27 cm, what is the natural length of the spring?
Any help would be appreciated!
To find the natural length of the spring, we need to understand the relationship between the work done on a spring and its displacement.
When work is done on a spring, it stores potential energy in the form of elastic potential energy. The amount of work done on a spring is equal to the change in elastic potential energy.
The elastic potential energy in a spring can be calculated using the formula:
PE = ½kx²
Where:
PE = elastic potential energy
k = spring constant
x = displacement from the natural length
In this case, the work done on the spring to stretch it from 15 cm to 21 cm is 54 J, and the work done to stretch it from 21 cm to 27 cm is 90 J.
Using the formula for elastic potential energy, we can equate the work done with the change in potential energy and solve for the spring constant (k).
For the stretch from 15 cm to 21 cm:
54 J = ½k(21 cm - 15 cm)²
54 J = ½k(6 cm)²
54 J = 18k
For the stretch from 21 cm to 27 cm:
90 J = ½k(27 cm - 21 cm)²
90 J = ½k(6 cm)²
90 J = 18k
From these equations, we can see that the spring constant (k) is the same in both cases, as the displacement and work done are the same.
Now that we have the spring constant, we can use it to find the natural length of the spring. The natural length is the position where the spring is at equilibrium, meaning no work is done on it.
When the spring is at equilibrium, the elastic potential energy is zero. Using the formula for elastic potential energy and setting it to zero, we can solve for the displacement (x) at equilibrium:
0 = ½kx²
Since the elastic potential energy is zero, the displacement (x) at equilibrium must be zero. Therefore, the natural length of the spring is the displacement at equilibrium, which is zero.
In conclusion, the natural length of the spring is 0 cm.